Extriangulated ideal quotients, with applications to cluster theory and gentle algebras

TL;DR

This paper generalizes ideal quotient constructions in extriangulated categories, applying them to cluster categories, Higgs categories, and gentle algebras, revealing new categorical equivalences and structures.

Contribution

It introduces a new interpretation of ideal quotients as extriangulated category quotients and applies this to various algebraic and categorical contexts.

Findings

Quotients are equivalent to homotopy categories of two-term complexes.

Extriangulated structures are well-behaved and 0-Auslander.

Applications include cluster categories and gentle algebras.

Abstract

We extend results of Br\"ustle-Yang on ideal quotients of 2-term subcategories of perfect derived categories of non-positive dg algebras to a relative setting. We find a new interpretation of such quotients: they appear as prototypical examples of a new construction of quotients of extriangulated categories by ideals generated by morphisms from injectives to projectives. We apply our results to Frobenius exact cluster categories and Higgs categories with suitable relative extriangulated structures, and to categories of walks related to gentle algebras. In all three cases, the extriangulated structures are well-behaved (they are 0-Auslander) and their quotients are equivalent to homotopy categories of two-term complexes of projectives over suitable finite-dimensional algebras.